A voltage `V` is applied to a `dc` electric motor. The armature winding resistance is equal to `R`. At what value of current flowing through the winding resistance is equal to `R`. At what value of current flowing through the winding will the useful power of the motor be the highest ? What is it equal to? What is the motor efficiency in this case ?

Explanation

To solve the problem, we need to find the current at which the useful power of the DC motor is maximized, the value of that maximum power, and the motor's efficiency at that point.

### Step-by-Step Solution:

1. **Understanding the Circuit**:
The total voltage \( V \) is applied across the armature resistance \( R \) and the load resistance \( R_0 \) (the motor's load). The total current \( I \) flowing through the circuit can be expressed using Ohm's Law:
\[
I = \frac{V}{R + R_0}
...